Theresa May will press ahead with a Brexit-dominated Queen’s speech shorn of a series of controversial social policies after failing to complete “confidence and supply” negotiations with the Democratic Unionist party.

The Conservative leader will become the first prime minister in decades to lay out a legislative programme without a guaranteed House of Commons majority after DUP sources said the two-party negotiations “haven’t proceeded in the way we would have expected”.

The Northern Irish party accused Downing Street negotiators of being chaotic and said the “Conservative high command ought to stop their backbenchers whingeing about the DUP and show our party some respect”. The DUP added that the party “can’t be taken for granted”.

No 10 sought to brush off the intervention, with May promising on Tuesday that she would use the Queen’s speech to respond with “humility and resolve to the message the electorate sent” after the snap general election left the Tories with 318 seats, eight short of an overall majority.

Plans for a new generation of grammar schools will be among a number of high-profile policies to be dropped by the PM, in a speech which will instead emphasise Brexit, counterterrorism and the least controversial aspects of the domestic agenda. The speech is likely to steer clear of any mention of contentious issues such as gay marriage or abortion.

Ian Lavery, the chair of the Labour party, accused May of having “no mandate, no authority and no programme” to even win support in parliament for those areas of policy.

“This is chaos from a weak and wobbly prime minister,” he said. “At the moment, it looks like the Tories are unable to put together even a basic programme for government in the Queen’s speech, or stitch together a stable deal with the DUP to stay in office.”

Tory sources insisted the government was confident of winning support for its legislative agenda, with an agreement with the DUP still expected to be signed off on Thursday – well before next week’s crucial parliamentary vote on the Queen’s speech. Defeat for May in that vote would mean her government would fall.

May’s own backbenchers – including some of her most vociferous critics – also made clear that they would back the government at this point, although made no promises for continued support in the medium term.

In order to secure enough parliamentary backing, May will have to scrap plans to end free school meals for younger pupils, and drop her party’s social care policy, including means-testing of the winter fuel allowance alongside lifting the ban on selective schools.

She insisted, however, that she would be pushing ahead with Brexit legislation, including a “great repeal bill – to bring EU regulations into UK law – and a series of other bills linked to customs, trade, immigration and agriculture.

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“Much has been said in recent days about what the general election signified about Britain’s decision to leave the EU,” she said, responding to a clamour from some for a softer form of Brexit given the new parliamentary mathematics.

“The fact is that over 80% of the electorate backed the two major parties, both of whom campaigned on manifestos that said we should honour the democratic decision of the British people.”

A cabinet minister said Jeremy Corbyn’s party was essentially in the same position as the Tories when it came to ending free movement and leaving the single market, despite a number of backbench Labour MPs questioning the position.

The shadow chancellor, John McDonnell, said Labour would seek to create alliances with Tory rebels and other parties to “reverse some of the most dangerous cuts” of austerity.

“There’s a whole range of issues around austerity where we think there may well be majorities in parliament now to reverse some of the proposals, particularly those brought forward in the last budget,” he told BBC Radio 4’s Today programme.

“We’ll finalise the drafting of amendments over next the next few days. We’ll wait to see the Queen’s speech today. From all the reports it’s pretty light.

“But remember, most of the austerity that’s rolling forwards is from the budget, and we won’t see another budget until the autumn.”

May said her government would be one that “consults and listens” as she struck a more conciliatory tone after the election seriously weakened her leadership and resulted in the loss of her two closest advisers, Nick Timothy and Fiona Hill.

“The election result was not the one I hoped for, but this government will respond with humility and resolve to the message the electorate sent,” May said. “We will work hard every day to gain the trust and confidence of the British people, making their priorities our priorities.”

She will face a tense atmosphere on Wednesday, with protesters planning a “Day of Rage” march on parliament in protest at the government’s handling of the Grenfell Tower tragedy.

Organisers have written on Facebook that they hope to “bring down the government” and “shut down London”.

A difficult few weeks for May led to her facing crowds chanting “coward” after initially failing to meet people affected by the fire.

As well as the series of Brexit bills, May is expected to respond to recent attacks in Manchester, London Bridge, Westminster and Finsbury Park with beefed-up counter-terrorism plans, including a focus on social media.

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There will also be some uncontroversial domestic moves including a civil liability bill to tackle a “rampant compensation culture” in which whiplash claims have spiked.

There will also be a financial guidance and claims bill to help consumers make effective decisions, and draft legislation on cracking down on domestic violence – particularly where children are involved – and pushing ahead with plans to ban letting fees for renters.

Sources said the Queen’s speech would also include a consultation on social care and a bill on mental health - but nothing on the NHS more widely.

In order to pass the legislation next week, May is almost certain to need the support of the 10 DUP MPs to give her a majority in the Commons.

Problems have emerged in the discussions between the Tories and the DUP, with unionist sources saying “backbiting from Tory backbenchers” against their party had stiffened their resolve to dig in.

They said complaints by Tory MPs about having to deal with the DUP had gone down badly. “They attempted to bounce us into a deal on Saturday night and then we had the backbenchers saying that we were unsavoury, etc.”

There is also a concern within the DUP that extra spending in Northern Ireland could trigger a barrage of complaints from English, Scottish and Welsh Tories over the special treatment.

The DUP wants to abolish air passenger duty for Northern Ireland’s airports – but there are concerns that a special deal could lead to demands for the same elsewhere in the UK.

The DUP deal is likely to be one of “confidence”, through which the party will back the Queen’s speech and support the government in any confidence votes, and “supply”, which refers to support for budgets and financial measures.

Hannah White, the director of research for the Institute for Government thinktank, explained how May would have to operate to survive the parliament.

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“Getting legislation through the Commons should be possible with effective whipping,” she said.

“While government and opposition numbers on public bill committees will be evenly balanced, procedural conventions around the way the chair votes when a committee vote is tied mean the Conservatives should be able to get bills through committees without any amendments.

“Then, drawing on the support of the DUP or other parties, they may be able to make any amendments they need at the report stage.”

May will, however, face fierce resistance to some of her Brexit legislation, with a parliament in which opponents of a hard Brexit feel emboldened by the general election result.

Frances O’Grady, the TUC general secretary, said: “Theresa May failed to win a mandate at the ballot box for her no-deal Brexit. Instead, we need a Brexit deal that puts jobs and rights at work first.”

She said the great repeal bill “must not be allowed to get away with eroding the rights that working people got from the EU”.